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Examples Of Inquiry

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Posted by: Jon Awbrey

EOI. Note 1

The question arises whether simple programs can
emulate the proceedings of scientific inquiries --
that is, to what extent is inquiry algorithmic?

I will approach this question through simple examples.

Let us first consider John Dewey's "Rainy Day Inquiry"
or "Sign of Rain" example, and this time put it under
the microscope and look at a few of its finer details.
In particular, I can use it to illustrate a couple of
important issues:

  1. The interpretive aspect of inquiry as a semiotic process.

  2. The differential aspect of inquiry as a dynamic process.

For ease of reference, I repeat here the original story:

| A man is walking on a warm day.
| The sky was clear the last time
| he observed it; but presently he
| notes, while occupied primarily with
| other things, that the air is cooler.
| It occurs to him that it is probably
| going to rain; looking up, he sees
| a dark cloud between him and the sun,
| and he then quickens his steps.
|
| What, if anything, in such a situation
| can be called thought? Neither the act
| of walking nor the noting of the cold is
| a thought. Walking is one direction of
| activity; looking and noting are other
| modes of activity. The likelihood that
| it will rain is, however, something
| 'suggested'. The pedestrian 'feels'
| the cold; he 'thinks of' clouds
| and a coming shower.
|
| John Dewey, 'How We Think', 1910, pp. 6-7

I will let this example soak in a bit
before I wring to my present purposes.

Jon Awbrey


Posted by: Jon Awbrey

EOI. Note 2

Susan Awbrey and I discussed Dewey's example of inquiry in our
article, "Interpretation as Action: The Risk of Inquiry", that
we gave at a Conference on "Hermeneutics and the Human Sciences"
in 1992, a revision of which was subsequently published in the
journal 'Inquiry: Critical Thinking Across the Disciplines',
Volume 15, No. 1, pages 40-52, (Autumn 1995). This paper is
available at the Arisbe Web Site or via the following links:

http://www.chss.montclair.edu/inqui...l95/awbrey.html
http://members.door.net/arisbe/menu...sp/aboutcsp.htm

Figure 1 indicates the "elementary sign relations" that are
involved in this fragment of inquiry. In particular, we have
the following two triples of the form <Object, Sign, Interpretant>:

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `o Sign (Cool Air, Dark Cloud)` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| Object (Rain) o------<| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `o Interpretant (Thought of Rain) |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 1. Sign Relation in Dewey's "Rainy Day Inquiry"

Here is what we said about the sign-theoretic aspects
of the inquiry process that we were able to detect in
Dewey's example:

| In this narrative we can identify the characters of
| the sign relation as follows: 'coolness' is a Sign of
| the Object 'rain', and the Interpretant is 'the thought
| of the rain's likelihood'. In his 1910 description of
| reflective thinking Dewey distinguishes two phases,
| "a state of perplexity, hesitation, doubt" and
| "an act of search or investigation" (Dewey 1991, 9),
| comprehensive stages which are further refined in his
| later model of inquiry. In this example, reflection
| is the act of the interpreter which establishes a fund
| of connections between the sensory shock of coolness
| and the objective danger of rain, by way of his
| impression that rain is likely. But reflection is
| more than irresponsible speculation. In reflection
| the interpreter acts to charge or defuse the thought
| of rain (the probability of rain in thought) by seeking
| other signs which this thought implies and evaluating
| the thought according to the results of this search.
|
| Awbrey & Awbrey, 1992

Next time I will take up the differential aspect
of inquiry as a dynamic process of theory change.

Jon Awbrey


Posted by: Jon Awbrey

EOI. Note 3

This time I will take up the differential aspect
of inquiry as a dynamic process of theory change.

Returning to the example in question, let me now draw
a more expansive picture of the "Rainy Day" situation,
one that augments the account with some consideration
of what our peripatetic/precipitate hero was probably
doing and thinking, however consciously or other-wise,
just slightly before the imaginary events in question.

It is clear that our hero did not begin his life
with the shock of cool air on his skin, at least,
not on this particular, late ambulatory occasion,
but probably had a prior distribution of default
beliefs about what this day was going to be like.

Just to extrapolate in a plausible vein of our imagination,
let us play along and say that the initial facts were thus:

Pulling the "conventional contingency" or "customary connection" trick,
let us then relativize the array of this data in the following fashion:

For the moment, let this figure of a "current situation" C
be one that is allowed to "go with the flow", letting its
letter be re-used to anchor whatever the case may become.

Now I do not know if it has to be the case that these three
features of the current situation had ever been entertained
by our ambler in any particular order, but we might suppose
their relative consistency as consisting in some such scene
as this: The hiker has formed a prior assumption about the
case that applies to the current situation, let's say, that
the day is balmy, C => B_1, an assumption that he will keep
as a default to continue in the same way until there arises
a reason to think otherwise. Further, we may imagine quite
plausibly that a rule of the form B_1 => A_1 can be applied
to the case C => B_1 to deduce the expectation of a certain
fact, namely, that the current situation will feature among
its phenomena the qualities of the air being warm, C => A_1.

So this logical set-up, or the likes of it, is what we may assume,
at least, plausibly enough for our currently illustrative purpose,
as the logical environ into which our soon to be inquiring ambler
strolls one fine and, for the moment, sunny day.

The rest you know. The cooler air, A_2, sensually contests
and logically contradicts the continuing assumption of the
prior condition A_1, demanding a fresh evaluation of the
conditioning assumption B_1, altering it into the new
hypothesis B_2, boding rain, which abduced case is
corroborated to a moderate degree by looking up
and spying a cloud in the sky, C_2.

Figure 2 manages to sum it all up in a fairly consummate fashion:

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `A_1` ` ` ` `A_2` ` ` ` ` ` ` ` ` ` `C_1` ` ` ` `C_2` ` |
| ` ` o~~~~>>>~~~~o ` ` ` ` ` ` ` ` ` ` ` o~~~~>>>~~~~o ` ` |
| ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` |
| ` ` ` \*` ` ` ` ` `*` ` ` ` ` ` ` ` `*` ` ` ` ` `*/ ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` |
| ` ` ` ` \ * ` ` ` ` ` * ` ` ` ` ` * ` ` ` ` ` * / ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` |
| ` ` ` ` ` \ `*` ` ` ` ` `*` ` `*` ` ` ` ` `*` / ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` * ` ` ` ` ` * ` ` ` ` ` * ` / ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` `*` ` `*` ` `*` ` `*` ` / ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ B_1 o~~~~>>>~~~~o B_2 / ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` `*` ` ` ` `*` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` * ` ` ` * ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `*` ` `*` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `\` * ` * `/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `\ * * /` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\*/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` C ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| A_1 `=` Air warm` ` ` ` ` ` ` ` ` ` ` A_2 `=` Air cool` ` |
| B_1 `=` Balmy day ` ` ` ` ` ` ` ` ` ` B_2 `=` Bodes rain` |
| C_1 `=` Clear sky ` ` ` ` ` ` ` ` ` ` C_2 `=` Cloudy sky` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` C `=` Current situation ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 2. Differential Signs of Rain

Jon Awbrey


Posted by: Jon Awbrey

EOI. Note 4

Here is a definition of what Peirce meant by a sign relation:

| A sign is something, 'A', which brings something, 'B',
| its 'interpretant' sign determined or created by it,
| into the same sort of correspondence with something, 'C',
| its 'object', as that in which itself stands to 'C'.
|
| C.S. Peirce, NEM 4, pp. 20-21.
|
| NEM 4 = 'The New Elements of Mathematics', Vol. 4,
| Edited by Carolyn Eisele, Mouton, The Hague, 1976.
|
| http://members.door.net/arisbe/menu...csp/l75/l75.htm

So when I say "coolness is a Sign of the Object rain, and
the Interpretant is the thought of the rain's likelihood",
it is because I think that the coolness in question brings
the thought of the rain's likelihood into the same sort of
correspondence with the objective event of rain as that in
which the coolness itself stands to the same event of rain.

Jon Awbrey


Posted by: Jon Awbrey

EOI. Note 5

For future reference -- if I may in fact refer
to a reference in the future -- here is a further
explanation of what Peirce meant by a sign relation:

| A 'Sign', or 'Representamen', is a First which stands
| in such a genuine triadic relation to a Second, called
| its 'Object', as to be capable of determining a Third,
| called its 'Interpretant', to assume the same triadic
| relation to its Object in which it stands itself to
| the same Object.
|
| The triadic relation is 'genuine', that is, its three members
| are bound together by it in a way that does not consist in any
| complexus of dyadic relations. That is the reason the Interpretant,
| or Third, cannot stand in a mere dyadic relation to the Object, but
| must stand in such a relation to it as the Representamen itself does.
|
| Nor can the triadic relation in which the Third stands be merely
| similar to that in which the First stands, for this would make the
| relation of the Third to the First a degenerate Secondness merely.
| The Third must indeed stand in such a relation, and thus must be
| capable of determining a Third of its own; but besides that, it
| must have a second triadic relation in which the Representamen,
| or rather the relation thereof to its Object, shall be its own
| (the Third's) Object, and must be capable of determining a Third
| to this relation. All this must equally be true of the Third's
| Third and so on endlessly; and this, and more, is involved in
| the familiar idea of a Sign; and as the term Representamen is
| here used, nothing more is implied.
|
| A 'Sign' is a Representamen with a mental Interpretant.
|
| Possibly there may be Representamens that are not Signs.
|
| Thus, if a sunflower, in turning towards the sun,
| becomes by that very act fully capable, without
| further condition, of reproducing a sunflower
| which turns in precisely corresponding ways
| toward the sun, and of doing so with the
| same reproductive power, the sunflower
| would become a Representamen of the sun.
|
| But 'thought' is the chief, if not
| the only, mode of representation.
|
| C.S. Peirce, "Syllabus" (c. 1902), 'Collected Papers', CP 2.274



Posted by: Jon Awbrey

EOI. Note 6

With this much background penciled in, let's revisit again
the contextualized picture or differential figure that we
derived from Dewey's "Sign of Rain" example:

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| `Air Warm ` `Air Cool ` ` ` ` ` ` `Clear Sky` Cloudy Sky` |
| ` `A_1` ` ` ` `A_2` ` ` ` ` ` ` ` ` ` `C_1` ` ` ` `C_2` ` |
| ` ` o~~~~>>>~~~~o ` ` ` ` ` ` ` ` ` ` ` o~~~~>>>~~~~o ` ` |
| ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` |
| ` ` ` \*` ` ` ` ` `*` ` ` ` ` ` ` ` `*` ` ` ` ` `*/ ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` |
| ` ` ` ` \ * ` ` ` ` ` * ` ` ` ` ` * ` ` ` ` ` * / ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` |
| ` ` ` ` ` \ `*` ` ` ` ` `*` ` `*` ` ` ` ` `*` / ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` * ` ` ` ` ` * ` ` ` ` ` * ` / ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` `*` ` `*` ` `*` ` `*` ` / ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\`Balmy` ` ` ` ` ` Boding`/` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ B_1 o~~~~>>>~~~~o B_2 / ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` `*` ` ` ` `*` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` * ` ` ` * ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `*` ` `*` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `\` * ` * `/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `\ * * /` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\*/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` C ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` Current Situation ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| A_1 `=` Air warm` ` ` ` ` ` ` ` ` ` ` A_2 `=` Air cool` ` |
| B_1 `=` Balmy day ` ` ` ` ` ` ` ` ` ` B_2 `=` Bodes rain` |
| C_1 `=` Clear sky ` ` ` ` ` ` ` ` ` ` C_2 `=` Cloudy sky` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 3. Signs of Rain Viewed in Their Natural Context

The question lingers: What justifies our calling by the name
of inquiry Dewey's sample of a semiotic process, namely, the
sign relational transit from the fact of cool air, C => A_2,
to the case of rain, C => B_2, and from the thought of that
case to the ruly act of looking up for any further signs
of rain that might be in the sky?

Let's try to focus for a while on what we may see as
the "differential" or the "distributional" aspect of
inquiry. If you follow the idea that inquiry begins
with a state of tension in the affected agent of the
process, then you are likely to recognize the legion
of diverse names for this annoyingly irritating mode
of being -- doubt, problem, surprise, uncertainty --
as forming variable manifestations of a differential
theme, for example, that a difference exists between
what an agent observes or accepts as actual and what
that agent either expects or intends to be the case.

Here, the agent has an initial expectation of fair weather,
due most likely to his initial observations of a clear sky,
but then discrepant sensations of significantly cooler air
cause him to pause, to reflect, and to update his forecast
of the imminent weather conditions to a foreboding of rain.

Jon Awbrey


Posted by: Jon Awbrey

EOI. Note 7

The question that drives me to examine these examples of inquiry
is the relationship among signs, information, inference, and the
typical trajectories of inquiry. At this point in the discussion,
we need a bit of background information about the pragmatic theory
of inquiry. I am presenting the long version of that on a another
thread, stemming from either one of these alternative urlocations:

INTRO. http://forum.wolframscience.com/sho...hp?threadid=598
INTRO. http://stderr.org/pipermail/inquiry...hread.html#1720

To keep from holding up this more concrete discussion of examples,
here is a short introduction to the principal ideas, as I see them:

It is frequently useful to approach the concept of an inquiry process
as a specialization of a sign relation, in the following three phases:

1. Sign Relation

A "sign relation" simpliciter, L c O x S x I, could be just about any
3-adic relation on the arbitrary domains O, S, I, so long as it
satisfies one of the adequate definitions of a sign relation.

2. Sign Process

A "sign process" is a sign relation plus a significant sense of transition.
This means that there is a definite, non-trivial sense in which a sign
determines its interpretant signs with respect to its objects.
We often find ourselves writing "<o, s, i>" as "<o, s, s'>"
in such cases, where the semiotic transition s ~> s'
takes place in respect of the object o.

3. Inquiry Process

An "inquiry process" is a sign process that has value-directed
transitions. This means that there is a property, a quality, or
a scalar value that can be associated with a sign in relation to
its objects, and that the transit from a sign to an interpretant
in regard to an object occurs in such a way that the value is
increased in the process. For example, semiotic actions like
inquiry and computation are directed in such a way as to
increase the "aptness", "brevity", or "clarity" of the
signs on which they operate.

Jon Awbrey


Posted by: Jon Awbrey

EOI. Note 8

Here's the "New List" text about the relations between
the types of signs and the types of inference, that is,
the morphological and temporal constituents of inquiry:

Let the expression "P_1 & P_2 & P_3 & P_4"

denote the proposition Q = Conjunction (P_1, P_2, P_3, P_4).

Then we may draw the following Figure of Abduction:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` P_1 ` P_2 ` ` ` ` P_3 ` P_4 ` ` ` ` ` |
| ` ` ` ` ` `o` ` `o` ` ` ` ` `o` ` `o` ` ` ` ` ` |
| ` ` ` ` ` ` \*` ` \ ` ` ` ` / ` `*/|` ` ` ` ` ` |
| ` ` ` ` ` ` `\`*` `\` ` ` `/` `*`/`|` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ `*` \ ` ` / `*` / `|` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` `* \` `/`*` `/` `|` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` `*\ /*` ` / ` `|` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `.` ` `Q` ` `.` ` `|` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `|` ` `|`*` `|` ` `|` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `|` ` `|` `*`|` ` `|` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `|` ` `|` ` `|` ` `|` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `|` ` `|` ` `|`*` `|` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `|` ` `|` ` `|` `*`|` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `.` ` `|` ` `.` ` `M` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` `|` ` / ` `*` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` `|` `/` `*` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ `|` / `*`Case ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\`|`/`*` `S=>M ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \|/*` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `S` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
Figure 4. Abduction of the Case S => M

Let the expression "S_1 v S_2 v S_3 v S_4"

denote the proposition L = Disjunction (S_1, S_2, S_3, S_4).

Then we may draw the following Figure of Induction:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `P` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` /|\*` ` Rule` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `/`|`\`*` M=>P` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / `|` \ `*` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` `|` `\` `*` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` / ` `|` ` \ ` `*` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `.` ` `|` ` `.` ` `M` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `|` ` `|` ` `|` `*`|` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `|` ` `|` ` `|`*` `|` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `|` ` `|` ` `|` ` `|` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `|` ` `|` `*`|` ` `|` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `|` ` `|`*` `|` ` `|` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `.` ` `L` ` `.` ` `|` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` / ` `*/ \*` ` \ ` `|` ` ` ` ` ` |
| ` ` ` ` ` ` ` `/` `*`/` `\`*` `\` `|` ` ` ` ` ` |
| ` ` ` ` ` ` ` / `*` / ` ` \ `*` \ `|` ` ` ` ` ` |
| ` ` ` ` ` ` `/`*` `/` ` ` `\` `*`\`|` ` ` ` ` ` |
| ` ` ` ` ` ` /*` ` / ` ` ` ` \ ` `*\|` ` ` ` ` ` |
| ` ` ` ` ` `o` ` `o` ` ` ` ` `o` ` `o` ` ` ` ` ` |
| ` ` ` ` ` S_1 ` S_2 ` ` ` ` S_3 ` S_4 ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
Figure 5. Induction to the Rule M => P

Reference:

| C.S. Peirce, "New List", CP 1.559, CE 2, p. 58.
|
| Charles Sanders Peirce, "On a New List of Categories" (1867),
|'Collected Papers' CP 1.545-567, 'Chronological Edition' CE 2, pp. 49-59.
|
| http://www.peirce.org/writings/p32.html
| http://members.door.net/arisbe/menu...st/nl-frame.htm

Jon Awbrey

---------------------------------------------------------------------------------


Posted by: Jon Awbrey

EOI. Note 9

Given the above background, concepts, and data, what
is the proper way of seeing the relationship between
the two trios that we have drawn for Dewey's example
of inquiry, specifically, the sign-theoretic 3-tuple
and the syllogistic 3-angle? (See Figures 6 and 7).

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `o Sign (Cool Air)` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| Object (Rain) o------<| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `o Interpretant (Thought of Rain) |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 6. Sign Relation in Dewey's "Rainy Day Inquiry"

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `Air Cool ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` A ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ^^` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | ` `\` Rule` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` |A` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | b ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | `d` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` Fact` | ` u ` ` o Before Rain ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | ` `c` `^` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | ` ` e / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | ` `/ `Case` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` |/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` C ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `Current Situation` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 7. Abducing a Case from a Fact and a Rule

An immediately obvious difference between the two Figures
is that the sign triple has the "Thought of Rain" whereas
the syllogistic triple has the object state "Before Rain".
Is this a significant difference between the two diagrams?

I will think on it ...

Jon Awbrey


Posted by: Jon Awbrey

EOI. Note 10

Cathy Legg's "Missing The Bus" Example

Here is another remarkably instructive example that can be tackled
moderately well by blocking it out as a "zeroth order theory (ZOT):

| I'm waiting for my morning bus and it doesn't arrive: Surprise.
| I then think -- in the past sometimes my bus hasn't arrived when
| it's a public holiday I've forgotten about : This case should be
| the same (induction), I then form the hypothesis that it is
| a public holiday (abduction).
|
| Cathy Legg, "Missing the Bus", as posted to the Peirce List:
|
| Subj: Re: Chomsky on Peirce on Abduction
| Date: Fri, 28 Apr 2000, 15:39:25 +1000 (EST)
| From: Cathy Legg <...>
| To: Peirce Discussion Forum <...>

Jon Awbrey

---------------------------------------------------------------------------------


Posted by: Jon Awbrey

EOI. Note 11

Cathy Legg's "Missing The Bus" Example (cont.)

Here is my analysis of Cathy Legg's "Missing the Bus" problem,
to the extent that it can be represented within the constraints
of propositional models, sentential logic, or zeroth order logic.

Let "C" represent the Current situation, that is,
the intinerant inquirer's current situation under
the circumstances of the problem in question, also
depicted by a "circle" in a venn diagram. This is
just a cheap propositional gimmick for covering to
some extent the "indexical" characterisitics of the
situation in question, but without having to resort
to the use of variables that range over domains of
"individual situations".

Next, let us contemplate the alternative possibilities,
formulated here as Proposition X versus Proposition Y.

As it happens, X is one's expectation, while Y is one's observation.
This difference between one's expectation and one's observation is
what one affectively experience as a surprise.

Let me stress this. The observed fact is Y, but what renders it
surprising is its difference from X, and this occurs on the point
of detaching the alternative consequents, A versus ~A.

Incidentally, it is this "differential" aspect of inquiry that led me,
starting about a decade ago, to begin to develop a "differential logic",
extending "propositional calculus" in almost precisely the same way that
differential calculus extends analytic geometry.

But let us get back to the situation at the bus stop.

The way that induction enters this situation
is as a component of previous cycles of inquiry
that led to the formation of a Rule, even if it is
only a "probable approximate rule", more or less as
formulated in Proposition K:

It does not affect the analysis at all if you have in mind another
sort of descriptor than "best", say, "normal", "ordinary", or so on,
so long as you acknowledge the conducive function or the mediating role
of any middle term like B.

When our traveller gets to the bus stop, it is most likely
that she is in a slightly confused, indeterminate, uncertain,
or vague state of mind, in the sense that she has probably not
even stopped to ask herself the question we'll call Question Q:

Consequently, she has walked, or ran, as is frequently the case,
right into the current situation, operating under the influence
of something like the following train of an automatic deduction:

And this is just where we came in, with the discrepancy between
the expected fact X : C => A and the observed fact Y : C => ~A.

The surprise that one meets with, instead of the bus,
might lead one to question all sorts of things. Any
number of speculations might come to mind. Among the
more rational possibilities, the surprise might cause
one to inquire into any and all of the premisses that
fed into the above deduction, if not the axioms of the
logic that one happens to be implementing at the moment.

But let's suppose that one lights on the Case C => B, as it is most
frequently the Case that is the cause of the problem, and therefore,
in accord with a higher order induction of the inquiry into inquiry,
it is most frequently the Case that empirical people consider first.

And so, after reflecting on the situation, and eliciting certain features
of how one's habitual reasoning processes fed into it, quasi modo intuitio,
one decides to vary the description of the Case, in this case, from saying
that C => B to asking whether it might not be true that C => ~B, that is,
asking oneself, "Can it be that the current situation is not actually the
best (modal, normal, ordinary, usual, ...) case, and that this may be the
cause of my expectation being disappointed?"

Jon Awbrey

---------------------------------------------------------------------------------


Posted by: Jon Awbrey

EOI. Note 12

Cathy Legg's "Missing The Bus" Example (cont.)

Let us now illustrate the particulars that we find
in "The Case of the Missing Bus" by using the sort
of "propositional logic in a lattice" diagram that
I used to articulate the basic brands of inference,
in what now must seem like so many long notes past.

Let me recap the story as we know it so far in the syllogistic
or "propositional constraint reasoning" (PCR) style of picture.

Figure 8 sketchily summarizes the first phase of the reconstruction.

o---------------------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `A` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` (A) ` |
| ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` |
| ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` |
| ` ` ` \ * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` * / ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` |
| ` ` ` ` \ ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` * ` / ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` |
| ` ` ` ` ` \ ` ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` * ` ` / ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` ` * ` ` ` ` ` ` ` ` ` ` ` ` * ` ` ` / ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` B o ` ` ` ` ` ` ` ` o (B) ` ` / ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` ` `*` ` ` ` ` ` ` `*` ` ` ` / ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` * ` ` ` ` ` ` * ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` `*` ` ` ` ` `*` ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ ` ` * ` ` ` ` * ` ` / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` \ ` `*` ` ` `*` ` / ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` * ` ` * ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` \ `*` `* `/ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` \ * * / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| A `=` Arriving bus situations ` \*/ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| B `=` Best case situations` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| C `=` Current situation ` ` ` ` `C` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o---------------------------------------------------------------------o
Figure 8. Cathy Legg's "Missing the Bus" Example

The point elements in these diagrams represent
the "propositions" that one is contemplating
with respect to a domain of objects, persons,
situations, and so on. Another option is to
treat them as the "terms" of the description:
Major term, Middle term, Minor term, and so on.

The line elements in these diagrams represent the "logical relations"
that are being considered between certain pairs of propositions, or else
the "premisses" that are being contemplated between various pairs of terms,
where roughly vertical lines indicate "implications", the antecedent lower
and the consequent higher, and where roughly horizontal placements indicate
relationship of "alteration" (change) or "alternation" (diversity), that is,
the situation among a number of alternatives, exclusive or inclusive, that
are available for one to change or to choose among.

It is my guess that something like this style of geometric figure
was used by Aristotle, and may have been a common sort of picture
at the time, at least, this is the impression that I get from the
way that he uses two different styles of language for indicating
the various sorts of logical relationships that are relevant to
the fundamental types of reasoning situation that he discusses.
For instance, Aristotle often uses the geometric label of the
line segment AB to indicate the premiss B => A. Of course,
this may just be a fluke of Greek grammar, or of its later
transcription.

There a convenient technical nomenclature that was added at a later date,
in which the various line elements depicting the premisses and relations
are customarily labeled as "Cases", "Facts", and "Rules", and I will use
this style of language rather freely to talk about the different roles
that different premisses may enjoy in the various forms of reasoning.

One other thing: I often use the following equivalent notations:

Among other things, this gives the following notational equality:

I hope that will be enough of a set-up to get this show on the road.

Data of the Situation:

  1. Alternative Facts: (C (A)) versus ( C ((A))), that is, (C A).

  2. Alternative Cases: (C (B)) versus ( C ((B))), that is, (C B).

  3. Alternative Rules: (B (A)) versus ((B)((A))), that is, (A (B)).

We meet the surprising Fact : C => (A), depicted by the line segment (A)C.
The reason that this Fact is surprising is that we automatically expected
a different Fact, namely, C => A. And, assuming the current situation C,
which we always do -- since this whole intervention of C is just a gimmick
for supplying a pivot to our thought -- we were led moreover to expect A,
the arrival of the bus.

If we stop to think about it, we come to realize that there is
a middle term that we have been taking for granted, say "B",
the "benign" situation, the "best case" scenario (assuming
that the best case means catching the bus), or maybe just
the modal, normal, ordinary, or usual case, if you like
those terms better.

The name "reflection" seems to fit the process by which
we can become aware of the previously automatic, implicit,
and probably unconscious deduction that led to a current
expectation, the one that is subject to conflict with
a current observation, thereby generating a dilemma,
a problem, or a surprise.

Nota Bene. Actually, I use the word "problem" more specifically
to refer to a difference between an intention and an observation,
but that is another, yet related story.

In the process of reflecting on the "program" of a habitual deduction,
we become able to identify the intermediate and the middle terms that
go "into it", and at this point we become able to contemplate their
deliberate variation. In this way, we become able to pass from the
class of propositions that are schematized by "B" to one or two in
the class of propositions that are summarized by "~B", and thereby
guessing a new Case, for example, that the current situation has
the marks of a public holiday, C => H, where H => ~B, and so is
not beneficial for our immediate purposes, tedious as they are.

Jon Awbrey


Posted by: Jon Awbrey

EOI. Note 13

Cathy Legg's "Missing The Bus" Example (concl.)

I left off last time at the point where you were just beginning to
contemplate the possibility that your current situation might fall
under the case description of a public holiday, thereby explaining
the absence of the expected bus, and a hypothesis which, if true,
would reduce your affective sense of surprise at the accustomed
bus not being there at the place-time that you were accustomed
to observe it.

Now, if you're like me, you might eventually think to look up,
and then to look around your surrounding neighborhood, to see
if you can observe any further evidence or any other naturally
occurring signs that might bear on your new hypothesis one way
or another.

This, of course, brings us to the deductive phase of our present inquiry.
And, equally of course, our immedately present phase of deduction must
be distinguished from all of those previous deductions, not to mention
their Promethean and Epimethean (fore and aft) bracketings by all of
those previous bits of abductive and inductive reasoning that went
into making up what were no doubt many previous cycles, and a vast
host of parallel cycles, and a countless array of epicycles on
our deference to an inquiry that may be indefinitely deferred.

Well, after that importunate word from our spontaneity,
I think that it is due time to get back to our story.
We have all been waiting for this bus long enough!

[This essay was written just after Easter 2000.]

If I had been walking on a residential street hereabouts,
through most of last week, when this "missing of the bus"
caper was alleged to have happened, I could have looked up
and looked around and seen all the gaily colored balloons,
the flapping ribbons, and the many other festive decorations
that were decked out on many of the houses and the trees by
all of the neighborhood parents who were throwing together
to treat their collective broods to an Easter Egg Hunt.
So that would have served to confirm the hypothesis of
a holiday, and perhaps it may have even altered my sense
of what was "best", "benign", "beneficial" -- trudging off
on my accustomed way, in pursuit of my habitual goals, or
stopping to enjoy the signs of another custom, and even
to follow them -- but that's another story altogether!

Anyway, it behooves me to try and size up the present moment of inquiry.
Let me unfold the map again and make a few additional notations upon it.

o---------------------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `A` ` ` ` ` ` ` ` ` ` ` ` `D` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` (A) ` |
| ` ` o ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` |
| ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` |
| ` ` ` \ * ` ` ` ` ` ` ` ` ` ` * ` ` ` ` ` ` ` ` ` ` ` ` ` * / ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` |
| ` ` ` ` \ ` * ` ` ` ` ` ` ` ` `*` ` ` ` ` ` ` ` ` ` ` * ` / ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` |
| ` ` ` ` ` \ ` ` * ` ` ` ` ` ` ` * ` ` ` ` ` ` ` ` * ` ` / ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` ` * ` ` ` ` ` `*` ` ` ` ` ` * ` ` ` / ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` B o ` ` ` ` * ` ` ` o (B) ` ` / ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` ` `*` ` ` ` `*` ` `*` ` ` ` / ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` * ` ` ` ` * ` * ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` `*` ` ` ` `*`*` ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ ` ` * ` ` ` ` o H ` / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` \ ` `*` ` ` `*` ` / ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` * ` ` * ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` \ `*` `* `/ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` \ * * / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| A `=` Arriving bus situations ` \*/ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| B `=` Best case situations` ` ` `o` ` ` D `=` Decorative situations |
| C `=` Current situation ` ` ` ` `C` ` ` H `=` Holiday situations` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o---------------------------------------------------------------------o
Figure 9. Cathy Legg's "Busman's Holiday" Example

I think that this pretty graphically says what I've been striving to say
in the last thousand words or so, and I am tempted to leave it at that,
but temptations to desist, you will have observed, are the sorts of
temptations I can easily resist! So let me attempt to sum it up
all over again, this time once again in schematic symbols and
in rather more verbose but slightly more descriptive phrases.

The validity of this abduction as a form of reasoning, in the only way
that its particular form of non-demonstrative inference can be said to
be valid, depends on the validity of the corresponding deduction, from
the Case : C => H and the Rule : H => ~A to the Fact : C => ~A. So it
needs to be remembered that the utility of this deduction, which only
concludes what has already been observed, is that it succeeds in its
aim to reduce the surprise of that observation.

The inductive phase, in this situation, consists of looking up and testing
whether the prediction comes true. I have been studying for few years now,
and still remain a bit puzzled, as to how exactly this sense of induction
fits in logically, if it does at all, with the other meaning of induction,
namely, of a non-demonstrative inference from a Case and a Fact to a Rule.

Jon Awbrey

---------------------------------------------------------------------------------


Posted by: Jon Awbrey

EOI. Note 14

Let's return to the question that I asked in EOI Note 9,
that had to do with the relationship between the semiotic
or sign-theoretic triad and the logical syzygy, for example:

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `o Sign (Cool Air)` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| Object (Rain) o------<| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `o Interpretant (Thought of Rain) |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 14.1 Sign Relation in Dewey's "Rainy Day Inquiry"

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `Air Cool ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` A ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ^^` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | ` `\` Rule` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` |A` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | b ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | `d` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` Fact` | ` u ` ` o Before Rain ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | ` `c` `^` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | ` ` e / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | ` `/ `Case` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` | / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` |/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` C ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `Current Situation` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 14.2 Abducing a Case from a Fact and a Rule

We need a word to cover all three uses of the later type of figure --
abductive, deductive, inductive -- since "syllogism" refers to the
deductive use alone, and so I will experiment with using the word
"syzygy" to cover all three ways of reading the same configuration.

Last time we looked at this situation I reflected as follows:

| An immediately obvious difference between the two Figures
| is that the sign triple has the "Thought of Rain" whereas
| the syllogistic triple has the object state "Before Rain".
| Is this a significant difference between the two diagrams?

For the sake of the NKS readers, there was some discussion of this point
on the Inquiry and Peirce Lists that they may find beneficial, and that
may yet come to fruition, but I will plod ahead on my own recognizance.

Here are the links to what record I was able to make of those discussions:

Peirce List:
http://lyris.acs.ttu.edu/cgi-bin/ly...?visit=peirce-l

Inquiry List:
http://stderr.org/pipermail/inquiry...hread.html#1707

Jon Awbrey


Posted by: Jon Awbrey

EOI. Note 15

In reviewing some of my previous writing on the issues in this area,
I came across the following collection of thoughts that seem of use.

For my part in this investigation, I have been
trying to resolve a couple of related problems:

  1. What is the proper articulation of the inquiry process in terms
    of the various kinds of inference, apodictic and approximate,
    that various thinkers have identified as being relevant to it?

  2. What is the proper placement of inquiry within a theory of signs?

My approach to this problem area has been to track back to the authors
of some of our initial ideas about signs and inquiry, to see if I could
work out for myself what they were thinking and how they moved from one
stage of their thought to the next, and maybe along the way to see if
I can see anything that they may have missed, or omitted to discuss
clearly enough. I am especially interested in the transition that
C.S. Peirce made from syllogistic to relational forms of thinking
about signs and inquiry, as that corresponds to an important task
in what might be called "computational architectronics", that of
building adequate logical systems on a solid propositional layer.

I have spent a fair amount of time staring at the likes
of the following two structures and trying to figure out
how they fit together, figuratively speaking, of course:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `o Sign ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` Object o---------O` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `o Interpretant ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
Figure 15.1 Elementary Sign Relation

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` Z ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` |\` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | \ ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | `\` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` \ ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` `\` Rule` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` \ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` `\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | Ab` > \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | `\ /` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` Fact` | <-o-De` o Y ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | `/ \` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | In` > / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` `/` Case` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` / ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` |/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` X ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
Figure 15.2 Three Kinds of Inference

After I had stared at the second picture a very long time,
I came to see that the two approximate forms of inference,
Abduction and Induction, have in common the property that
they bring a middle term into the immediate configuration.
Then I remembered that Aristotle is supposed to have said:
The essence of quick wit lies in grasping the middle term.

But where do these middle terms come from, anyway? It is
conventional to say that they come in with the abductions
of the cases that first evidence any need to call on them,
and that this is what puts them in the pot for inductions
and deductions to bid for them on any subsequent occasion.
But maybe it would make sense to recognize an independent
process, solely dedicated to finding or making mediations.
Conceived in this way, this process would be a duction in
the opposite direction from Deduction, dub it "Adduction".

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` Z ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` |\` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | \ ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | `\` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` \ ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` `\` Rule` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` \ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` `\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` Fact` | Ad ---> o Y ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` `/` Case` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` / ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` |/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` X ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
Figure 15.3 Adduction of a Middle Term

I'm not too committed to this name for the action,
and it has been used on one or two rare occasions
as yet another name for abduction, but I will use
it until I come up with a name that I like better.

Jon Awbrey

---------------------------------------------------------------------------------


Posted by: Jon Awbrey

Further discussion, links, and references on the logic of inquiry can be found in this article.


Posted by: danil

Nice info Jon now I know more about inquiries on proper way.

Daniel,
UNIQUE ARTICLE WIZARD




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